LINEAR ALGEBRA AND ITS APPLICATIONS
ELSEVIER
Linear Algebra and its Applications 283 (1998) 6173
Ergodic decomposition of Markov chains C6sar E. Villarreal 2 Departamento de Matenuiticas. Cent. de Investigachm y de Estudios. A vanzados del !. P. N., Apartado Postal 14740, 07000 M~xico D.F., Mexico
Received 14 October 1997; received in revised form 20 April 1998: accepted 23 April 1998 Submitted by R.A. Brualdi
Abstract We explicitly find the spectral decomposition, when it exists, of a Markov operator p. :fl ~ El using the asymptotic periodicity of the associated infinite Markov matrix. We give a simple condition under which an infinite Markov matrix is asymptotically periodic. We also determine the set of P'invariant distributions in t?i and the set of P'ergodic distributions. © 1998 Elsevier Science inc. All rights reserved.
I. Introduction Lasota et al. [I] prove a spectral decomposition theorem for a class of Markov operators T, called strongly constrictive, acting on an arbitrary space L I (X, ~', It) with a afinite measure/~. For these operators all the sequences (T"f), with f E L I, are asymptotically periodic. The result by Lasota et al. was extended b) ~Komornik [2] to the case of a weakly constrictive Markov operator. In this paper we give a method to explicitly find the spectral decomposition of a Markov operator P* : f~ ~/?~. The method is similar to the one given in [3] for finite Markov matrices, which is based on results by Chi [4]. In Section 2 we state the basic definitions. In Section 3 we prove some results on idempotent infinite Markov matrices, which are needed to explicitly
I This work was partially supported by CONACyT grant 3115PE9608. 2 Email:
[email protected] 00243795/98/$19.00 © 1998 Elsevier Science Inc. All rights reserved. PiI: S 0 0 2 4  3 7 9 5 ( 9 8 ) i 00800
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C.E. Viflarreai I Lim, ar Algebra and its Applications 283 f 1998) 6173
find the spectral decomposition given by the Spectral Decomposition Theorem (SDT) of Section 4 for a M a r k o v operator P* : t?t + P. We provide a condition for P" to be constrictive and a method is given to determine the number of orthogonal vectors in the SDT. In Section 5 we characterize the P*invariant distributions and the P*ergodic distributions.
2. Preliminaries Definition 1. An infinite Markov (or stochastic) matrix is an infinite matrix 3~ P = (P~J)~j=I with nonnegative components P0 such that the sum of the entries of each row is 1, that is, ~"~y~=lpij = 1 for all i = I, 2 , . . . Let/?l be the Banach space of sequences II = (xi,x=,...) in I~, seen as row vectors, such that Ill, ll, " ~ L , Ix, I < ~ . The convex set D := {t~ e e': I1~11, = 1. and ;t 1> 0} is referred to as the set of distributions in e ~. Definition 2. A positive linear operator P* : ,fi _.., ~,l is called a Markov operator on ~l if it maps D into itself, that is,
p ' ( o ) c o. In particular, an infinite Markov matrix P defines a Markov operator P* on fl as
P'(~) := i~P,
(I)
so that t h e j t h component of P'(/~) is (/tP)i = ~_,k~ i x~pk,. Also note that P" is a contraction map, that is,
llP'(i~)ll, <~ lll~ll,
v~ e t'
(2)
and, moreover, P" preserves the norm of 1~ E t '~ if l~ is nonnegative, i.e.,
llP'(i,)ll, = llt~ll,
vl~ e .e'~.
(3)
Throughout the following, P = (P,),,il denotes a given (infinite) Markov matrix, and P' stands for the corresponding Markov operator defined by Eq. (I). Further, I~1 denotes the set of positive integers, and if B is a subset of I~, we define
P(i,B) "= ~ ,
for i E I%1.
As in Markov chains theory we interpret P(i,B) as the "probability" of going from i to the set B in one time unit. "x. We will identify a sequence l~ = (x;)i:~ ! E e ~ with the finite signed measure (also denoted by l~) on the measurable space (1~,2 ~) such that It(B) := ~,~a.~:, for B C ~.
C.E. Villarreal I Linear Algebra and its Applications 283 (1998) 6173
63
Let us recall some definitions (see, for instance, [5]). Definition 3. (a) A set B C I~ is called Phlvariant if P(i,B) = I for all i E B. (b) A distribution It E D is called P*invariant if P* (1~) = l~(c) A distribution It E D is said to be P*ergodic if it is P*invariant and ll(B) = 0 or 1 for any Pinvariant set B.
In Section 5 we give conditions to identify the set ff~ of P*invariant distributions and the subset De~: c Dt~ of P*ergodic distributions. '~ .'X. Definition 4. Two sequences p = (X i)i=l and v   . . (Y,)~:l in ~! a r e said to be orthogonal if 1~" v := ~'~i~i xiyi = O. in this case, we write II _1_ v.
3. Idempotent infinite Markov matrices
The main result in this section is Theorem 10, which requires some properties of idempotent infinite Markov matrices. These properties, stated in the following iemmas, are also used in the next section. (Recall that a matrix A is said to be idempotent if A' = A.) Lemma 5. Let A = (a,:i)cj= ! bc an idempotem it~'nite M a r k o v matrix. ( a ) / f akk = 0 fi~r some k E [~, then aik = O.[or all i E ~. (b) For all k, i E [~, we have ai~ <~akk. Proof. (a) Arguing by contradiction, suppose that akk = 0 and a,k > 0 for some i E N . Let i t = s u p { a i k : i E N } . For each ~ : > 0 we take i , : E N such that / i  ai, k < ~: and a,',k > 0. Then by the idempotency hypothesis,
{j:
1i: a/~ >0}
j::: I
{j: ,ajk>0}
and, therefore,
':I
ai, k
a,0j t> !    . {j: a,~ >0}
a/A ::>0}
(4)
ai,~.
Now, as ark = 0, we have that k ¢~ {j: ai~ > 0}. On the other hand, a;,k > 0 yields !j: a,~ >0}
{.i: a,k .:,0}
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CE. Villarreal I Linear Algebra and its Applications 283 (1998) 6173
Let us now consider a sequence ( e , ) ~ of positive numbers such that lim~_.,~e., = 0 and, furthermore, the limit lim~_~ ~0:a,~>0} a~.j = ' Q exists. Then, from Eqs. (4) and (5),
l>~ai~.,+
~,
a,~,j>
{j: ail >0}
Hence, since a 
~
a,.:.j>~ 1  e_..y_~.
(6)
ai,.,,k
U: aik >0}
~ < a~.k ~<~, letting n ~ oc in Eq. (6) we obtain
which is impossible because ~ > 0. This contradiction yields that we must have a~ = 0 for all i. (b) If a~k = 0, then a,k <~au. Now, if a,k is positive, then, by (a), so is au. Let be as in the previous proof, that is, a := sup {a~k: i E ~ 1, and for 0 < r. ~<~ let i,: ~ r~ be such that ~  a;,t < ~. As A is idempotent, we have
j=l
j:l
Now, in Eq. (7), thejth term of the first sum is less than or equal to thejth term of the second and, further, the difference between these terms is at most ~. Thus if a,,j > 0, then a,,:ja/t + ~: > a,,ja,.t, so that aj~ + ~:/a,,:j > a,,~. In particular, a~+~>a~,, ai, k
(8)
>~~:.
Finally, in Eq. (8) we take the limit as ~: . 0 and we get at, I> a, so that, by definition of a, we have a,~ <~a~t for all k, i ~ l~l. lI Lemma 6. Let A = (au)~4:=t be an idempotent it#'nite Markov matrix, l./'a,k > 0 ./'or all k E ~ and ~"~2~:!akk < oc. then: (a) aj, > 0 =~ a 0 > 0 (or, equivak, ntly, a u = 0 ~ ay~ = 0). (b) a~k > 0 =~ a,k = akl.
Proof. (a) By Lemma 5(b),
Sa" ESo,,,,, i I
t::l /=1
Sa,,Eo,, Sa,, Ea,. i=1 i::1
i::l
t:l
1:::1
i::1
Thus. as a,, < ~ . we obtain a , a , = aria .. Therefore, ai, > 0 implies a , = a,. Hence. as a , > 0 (by hypothesis), we conclude that ai, > 0 implies a o > O.
(b) By Lemma 5(b),
i=1
i=1
C E. Viilarreal i Lim'ar Algebra and its Applications 283 (1998) 6173
Therefore. ak~ > 0 implies a~ = aa, so that, by (a), a;k > 0 implies a,k = akk.
65
iq
Remark 7. If A = (a~j)~=~ is an idempotent infinite Markov matrix such that all the entries of the columns k , , k 2 , . . , are zero, then the matrix obtained by cancelling all the rows k , , k 2 , . . , and all the columns k~,k2,.., is also an (infinite or possibly finite) idempotent Markov matrix. Remark 8. T w o sequences II = (xi)i~! and v = (v~)i~=l in D are orthogonal if and only if their supports are disjoint, that is, x~ > 0 implies yi = O, and yi > 0 implies x~ = O.
We will denote by a,. the ith row of A, that is, a;.= (au)j: i . Lemma 9. Let A = (au)c~: ! be an Mempotent infinite Markov matrix with a,i > 0 f o r all i E ~l and ~']i~l aii < o¢. Then any two rows o f A are either equal or orthogonal; that is, Jbr all i, k E ~ we have ai. = ak. or ai. .L ak. Proof. Suppose that the rows ai. and ak. of A are neither equal nor orthogonal. Let: B = { j ~ , j 2 , . . . } be the set of all indices such that au,, = akj~ > 0 for all jp ~ B; B* = {j~,j.~,...} be the set of all indices such that au; > 0 = akj;, for all Jp E B*~, B** = {JI ,J2 ~,'"} be the set of all indices such that a~j;,. = 0 < aki;,, for all j~,*E B**; and B = {~l,j.,,...} be the set of all indices such that a0,, = ak),, = 0 for all Jr E B. As A is a Markov matrix and the vectors a,. and ak. are neither equal nor orthogonal, from Lemma 6(b) and Remark 8 we have B :~ 0, B * # 0 and B'" ~ 0. By definition, 0 = au:,.= E a i t , avj7 = Eau,,a,,,,:,.. p= !
Thus aj,,j:. = 0
it,CB
for all j , ~ B and for all jq" ~ B*'.
(9)
Now, if a,, > 0, by Lemma 6(b) we have
arA, ~~ Eat.paf~  at:~. E p= i
arpo
{p:,~, #o}
Therefore, )~.{s,:,,,,,~01a,v= I and, as A is a Markov matrix, (a,., > 0 and a~ = 0)
=~ a, v = 0.
(10)
By Eq. (10), and because au; > 0 = a~ji, we have a~k = 0; that is, k is in the union B*" U/}. We will next show that tins leads to a contradiction. Indeed, suppose that k ~/}. In this case, akk = 0, which contradicts our hypothesis; therefore, k e B*'. But, from Eq. (9), aj, k = 0; thus, by Lemma 6(a), akj, = 0,
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C.E. Villarreal I Linear Algebra and its Applications 283 (1998) 6173
which contradicts that j~ E B. Therefore, the rows a~. and ak. are either equal or orthogonal. I1 Theorem 10. Let A = (aij)ij=t be an idempotent infinite Markov matrix such that E~=i a~ < ¢x~. Then: (a) Any two rows with positive diagonal entries are either equal or orthogonal; that is, JOT all i, k E [~ such that a~ > 0 and akk > 0 we have a~. = a,. or a~. 3. ak.. (b) Each row o f A is a convex combination o f the rows which have a positive diagonal entry.
Proof. (a) This part follows from Lemma 9 and R e m a r k 7. (b) Let ak. be the kth row of A for a fixed positive integer k. To prove (b) we wish to write ak. as a convex combination a~. = ~t(ml,k)am.. + ~(m2,k)am,.. + . .  ,
where an,..,an,:.,.., are rows with positive diagonal entry. We will first show that if am. is a row such that amy,a,.t and atom are all positive; then: (i) The rows aj., ai. and a,.. are equal. (ii) If a,j = ajj for some i, then a,! > 0; and similarly, if a,t = all for some i, then a~/> 0. (iii) (i: a,, ::a,I
{i: a,t ::,,)
(iv) a~ = ~(m, k)a,,i
and
akl = =(m, k)a,,i.
Proof of (i). By Lemma 5 and part (a), the rows aJ., at. and am. are equal. Proof of (i). It is impossible to have a~ = ajj and aa = 0 because this would imply, by idempotency, that aoajt = 0 with a~j > 0. Therefore art = 0 . However, by Theorem 5(b), an,ajj > 0, so, by (a), the rows al. and a t. would be orthogonal, which contradicts that they are equal. Hence a;j = aJJ for some i implies a~t > 0. Similarly, a~l = an for some i implies a~j > 0.
Proof of (iii). By (a) and Lemma 5(a), (0 < a , < all or 0 < a~j < aJJ) ~ a, = 0 =~ ak, = 0. From this fact and (ii) we have that if a,~ = ajj and aa < all, or a ~ j < a j j and a , l = a l l : then ak~=0. Hence { i : a , j = a i i and a k , > 0 } = {i: ail= alt and a~ > 0}, and, therefore,
a/. = which proves (iii).
~
aki,
C.E. Villarreal { Linear Algebra and its Applications 283 (1998) 6173
67
Proof of (iv). In the proof of (iii) we have that 0 < aij < ajj ~ aki = O. Thus, by (i),
akj=Zakiaq=
Z
i= i
akiaij=ayj Z
{i:aq=a H }
 a,,,j Z
aki
{i:aq =aj; }
ak' = cx(m'kla'J'
{ i:aq =at i }
and similarly akt = ot(m,k)aml,
which proves (iv). Note that if two rows am., and a,,. with positive diagonal entry are equal, then the coefficients a(m,k) and ~(n,k) are equal. Now, having (i) to (iv), we can easily coml~!ete the proof of part (b). Let us define m l , m 2 , . . . E • recursively as follows: the integer ml is such that am~.is the first row of A with positive diagonal entry. Next, m2 is the smallest integer greater than ml such that the row a,,:. has a positive diagonal entry and am2. # a,,,.. Continuing this process we obtain a (possibly finite) sequence (ml,m2,...) in which, given mqi, we choos,: mq as the smallest integer greater than mq_l such that the row am~. has positive diagonal entry and a,,~. # am,. for all i 1, ."~, . o . , q  1 Let r #{mj,m2, } be the number ofelements in the set {rot, m2,...}, so that r E t~ U {oo}. (In Theorem 19, we give an explicit value for r.) By construction, for each row a~. such that a,i > 0 there exists a unique m,! E {mr,m2,...} such that a,. =a,,,,.. If a,,~ = 0 , then from (iv), the definition of mi,m2,..., part (a) and Remark 8, we conclude that =
.
=
. . .
r
ak.
y]
mq, k )a,, . .
q=I r
Thus, as A is a Markov matrix, we have ~q=l ot(mq, k) = I with cX(mq,k) >t 0 and the proof of the theorem is complete. I1
4. The spectral decomposition theorem In this section we first state without proof a particular case of the SDT given in [2,6,7,1,8], and then we describe a procedure to determine the different components that appear in the spectral decomposition of a Markov operator. The main assumption in the SDT is the constrictivity of the Markov operator P': e i , ~t, which is defined as follows.
C.E. Villarreal I Linear Algebra and its Applications 283 (1998) 6173
68
Definition 11. We say that P*' f~ set F C ~ such thai
limsupd(P*"(lO,F) = 0
~ is constrictive if there exists a compact
['or all/a E D,
/'1"* ~
where d ( v , F ) : = i n f { l l v  Pil,: p ~ F}. In Theorem 17 we give a simple condition under which a Markov operator is constrictive. Moreover, Theorem 18 characterizes the set F0 = { v~, . . . . v,.} given in the SDT and, finally, Theorem 19 gives a formula to find the number of elements in F0. Definition 12. We say that l' E D is such that P*"(/t) = II.
P'periodic
if there is a positive integer n
Theorem 13 (SDT). Let P* be a constrictive Markov operator on fl. Then: (a) There exists • a finite set Fo = { v l , . . . , Vr} ofpairwise orthogonal P*periodic elements q f D, ® a set o f continuous linear fimctionals 21,..., 2r on ~l, and ® a permutation a o f the integers 1. . . . . r such that (I) lim lle'"(lO  ~7=, = o.Io,, each It 6_ g l ti ~t 7X, (11) P'(vi) = v.!,l j o t i = I . . . . . r. (b) The Jimc~kmals )., are positive, that is. ,;.,(1~) >f 0/fl~ i> 9. Moreover. r
L z,(v)= t
!
.torriD,
I
(111) I),,(l,)l ~ Ill, ll, .for 1' ~ t;'. (c) The sets { v l , . . . , Vr} and {,;.I, . . . . ,;.,.} sati.sfl,#lg (I) and (!I) are unique. Remark 14. If P = ( P u . ~x~ . i : l is an infinite Markov matrix. 7X. /6j :: sup {Pij" i E I%1}and b := ~/)j=l. Further, recall that P*" fl for the Markov operator defined by P, i.e., P*(I') ltP.
we dehne t~ stands
=
In the remainder of this section we suppose the following: Assumption 15. P is such that
Ilpll
< ~.
Remark 16. If Ab and the rows of P are seen as afinite measures on the measurable space (1~.2;"~), then Assumption 15 gives that/5 is tight (cf. [2]). Moreover, from Theorem 3.2 in [9] it follows that the family of rows of P is also tight. Theorem 17. Suppose that Assumption 15 is sati.~[ied. Then the matrix P &,fines a constrictive Markov operator P* on (,I.
C E. Villarreal t Lhu, ar AIgehra and its Applications 283 (1998) 6173
69
Proof. By hypothesis,/~ is in Ct. If It = (x~,x2,...) is in D, then P*(/,) ~
{t,'(~,). t, ~ 19} c ~ := {,, ~/9: ,.~
I, (~)~. " ~ t is " a subsequence k t n J  ~Jk= ~ such that 0,,lj)~_~ converges to a nonnegative n u m b e r yj. Let v = (v~,y_,,...), and observe that v E K and 0 , , ] i ) ~ converges to yi for each j. Given r. > 0, let N E r~ be such that ~~j~:x~ ~ ~2/~ < e. Now !
7"K,:
7X.;
lim IIv,,~  vii, = kl....i mx LX, , , ,'~l. v~,t , i
= lim
(
vii
.,
!

),,)/  v; + Z
i N
b,,:j  y,I +
)
i
.= ~'.
As ~: > 0 was arbitrary, we have lim~ ...., Iv,,,  vj~  0; in other words, the sequence (v,,),," I has a subsequence (v,,,)~l convergent in el to some v 6 K. Therefore, K is sequentially compact, whtch proves that K is compact in g~. Hence P* is a constrictive operator on g~. !1 /~
.
Theorem 18. Suppose that Assumpt.km 15 hoMs, and let r be the positive #tteger given m the SDT. Then there exists an #![inite idempotent Markov matrix A such that (P*("!")(#)),~l converges to A*(#) 01 gl ./or all # E[I. Moreover, the e&ments vl , . . . , v,. given Or the SD T are the rows o f A which have a positive diagonal ento,. Proof. By Theorem 17, there exists a finite set F0 = { v l , . . . , v,.} of pairwise orthogonal periodic elements of D and a set of continuous linear functionals { 2 t , . . . , 2,.} on ~,l that satisfy the condition of the SDT; in particular, r
lira I P*"(#)  ~~,~(~,)*'~,,¢~,llj  0 II "
k:l
In addition, since o"~"(k) = k, we have
for each il ~ [i.
70
CE.
Villarreal I Linear Algebra and its Applications 283 (1998) 6173 /,
k=l
where 1 if/=j, 0 if i ¢ j.
/i,j:=
Let yk, be the ith entry of v~ and let A = (ao)~j= ~ be the infinite matrix with rows r
a,. = lira 6~.P"!" = lim P""!"/(6~.)= ~21,(6,.)vk, tl~
7K.
/I'* "N.
(ll)
k=i
where, by the SDT, ~=l).k(6,.) = I and 0 ~ 2k(6~.) <~ i. Moreover, note that ,'k = P'{~'"'(,'k)= ~_rvk, lim P"~"'(6,.) = ~_rvk, a,. • i=l
(12)
i=i
Observe that A is idempotent and Markov, and ~~_l akk <~ )'~i/~ < o~. Therefore, by Theorem 10(b)and (12), the elements v l , . . . , Vr are convex combinations of the rows of A which have a positive diagonal entry. Hence, by Eq. (11) and part (c) of the SDT, v l , . . . , v~ are the rows of A with positive diagonal entry. Li An argument given in Ref. [7], p. 753, Eq. (I.5) shows that 0 < r
IlPll,,
which gives an estimate of the number r of distributions vl,..., v,. in the SDT. This is important 0ecause, even if we do not know r, it allows us to calculate A'(#) as the limit iimlIP't~!"l(/,) A'( 011, = 0
for any integer s >>,r,
n , ~,
in particular for s >1 IIpll,, The following theorem gives the precise value of r. Theorem 19. Suppose that Assumption 15 holds and let the positive integer r and ,4  ( 0)cj:t be as in Theorem 18. Then r is given br at
r
~
~altk. k=:l
Proof. By Theorem 18, v i , . . . , vr are the rows of A that have a positive diagonal entry, and these vectors are pairwise orthogonal, by Theorem 10(a). Let
71
C E. Viilarreal I Linear Algebra and its Applications 283 (1998) 6173
for l<~n<~r.
T,:={iE~'ai.=v,}
Then, from Theorem 10 and Lemma 5, r
k:l
n=l
{kETn}
r
r
n=i
k=!
5. P*invariant distributions
Let P*:e ~ ~ ~ be a constrictive Markov operator, and let D~, := I"I,,~_tP'"(D) be the set of all the limit points of the sequences (P*~(/z)),~ with It E D. By the SDT, v is in D:~ ifand only ifit is a convex combination of the distributions v~,...,v~. That is, D~ is the convex hull of {vl,..., Vr}. We will now identify the set D t c~ D~ . of P*invariant distributions and the set DE.~that consists of all the P*ergodic distributions (see Definiton 3). Two integers i and j in { I , . . . , r} are said to be equivalent (denoted by i ~ j) if P'k(v,) = vi for some positive integer k. Observe that ~ is an equivalence relation, and denote by O~,O?.,...,Od the different equivalence classes of { l , . . . , r } . Let Oi := {v~:i E Or}. F o r j = I , . . . , d , let 1 r~ "= # O j Evi__ •
(13)
ic=OI
be the "average" of the element:~ in Oj. Observe that P " O j ~ O; is bijectivc and that P'(v,) E O i ~ v, E 0 i. Therefore,
zv : JEOi
Cz,) ~'EO/
I'(O t
I(:OI
~kiEO;
which gives that zj is a P*invariant distribution. Note that r~,..., z,~ are mutually orthogonal. The proof of the following theorem is similar to the proof in Ref. [3], Theorem 10. Theorem 20 (Ergodic Decomposition Theorem). Let P*: E! , ~1 be a constrictive Markov operator de[ined b;, an infinite Markov matrix P and let D"x C D~ be the set o f all the P*invariant distributions. Then Dt.,c is a convex se,r anti, in fact. it is the convex hull t f { r I , . . . , *,t} with r i as in Eq. (i 3), that is, d
d
D~ = {1, E EI" t' = E ~ , z, with ~j >i 0 and E e i = j:=!
1}.
(14)
j=l
Hence, the colk'ction o f all the P'ergodic distributions is DE~ = { r l , . . . , Zd}.
C E. Viilarreal i Linear Algebra and its Applications 283 ¢1998) 6173
72
Proof. Let C be the convex hull of { r i , . . . , r d } . Since ry is a P*invariant distribution for j = 1,2,... ,d, any convex combination d
It
~OtjZj j=l
is also a P*invariant distribution. Hence C c / Y ~ . To prove that/Y~ c C, first note that D{~ c D,~ so that if p <5 D is a P*invariant distribution, then, by the SDT, l~ is a convex combination of vz,..., v, that is, r
(15)
1' = ~'~Jl,,,v,,,. m=

Also note that Eq. (15) is the unique representation of p as a linear combination of v~, v.,. . . . , v,, because these distributions are mutually orthogonal. Now, if i,k are bo~h in Oj, then the coefficients //, and fl, are equal. Indeed, if i, k E Oj, then there is a nonnegative integer t, such that v, = p,t(v~), and so r
l, :
P"(I,)
=
iI
r
~"~fl,,P"(v,,) = ZflmP"(V,,, ) + fl, Vk + Z/I,,,P*'(Vm). m.:l
m
m::::i+l
I
Hence, as the representation in Eq. (I 5) is unique, we most have fli = fl~. Now, for j = !, 2 . . . . , d, let si be an integer in Oj. Then io
m =
I
i(() I
t~:() 2
I!,, ( #0. )*. +/k,(#O2),.,
o: (),t
+ . . . + fl.,,,(#O,~),,~.
Finally, if for j = !, 2, . . . . d we take ~, = II,, (#Oi), then we get d .! i
and, moreover, ~ti >f 0 and ~, E ~ , fl.,,(#O,) E']:~, E,~o, lI, = ~ , ! fl,,, = I. Therefore, D I"x. C C, which completes the proof of Eq. (14). This in turn yields the last statement in the theorem, /yr~ = {rl,..., Zd}; see for instance, Kifer [5], Theorem i.I in Appendix A.I. [2] =
=
Acknowledgements The author is grateful to Professor Onesimo HernfindezLerma for his useful comments and invaluable suggestions concerning this paper.
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